No. Players fire one shot each in strict turn, whether it hits or misses. Azure fires first, and the turn always passes after your shot.
Naval Battle has no bonus shots. You fire one shot, you see whether it was a hit or a miss, and then it is your opponent’s turn, every time.
Who goes first
Once both fleets are placed, Azure fires first and the two players then alternate for the rest of the game.
What it changes
Because a hit does not earn you another go, every shot costs the same turn. That makes searching efficiently matter: a pattern that finds ships in fewer shots wins the race to 17 hits (how to win). The Naval Battle strategy guide explains how to search, and the full rules cover the rest.